Truth-Functors

Consider
"j and y".

If we had a sentence
consisting of this together with two constituent sentences, we would be
able to tell the truth-value of the sentence as a whole given simply the
truth-values of the constituent sentences. We would know that the whole
sentence would be true if both the constituent sentences were true, and
otherwise would be false. We could represent this fact about "j
and y" by the following
diagram:

jy

j
and y

T
T

T

T
F

F

F
T

F

F
F

F

A
diagram of this sort is called a truth-table. So, a truth-table
for a sentence-functor is a diagram which shows (in the case of a sentence
consisting of it together with constituent sentences) what value the sentence
as a whole will take for every possible combination of truth-values in
the constituent sentences.

We
can define a truth-functor thus:

A truth-functor
is a sentence-functor such that the truth-value of any sentence consisting
of it together with constituent sentences is determined solely by the
truth-values of the constituent sentences (together, of course, with the
meaning of the sentence-functor).

Alternatively,
we could define a truth-functor as a sentence-functor with a truth-table
- but the usefulness of this depends, of course, on the fact that we have
already defined truth-table.

Some sentence-functors
are not truth-functors: "j
because y", for example.
This is because, although the falsehood of either of the constituents
is enough to determine the truth-value of the sentence as a whole (it
must be false), the truth of both constituents is not enough to determine
the truth-value as a whole. For instance "Bill Clinton was president
of the USA because he secured most votes in the electoral college"
is true. But "Bill Clinton was president of the USA because the battle
of Hastings took place in 1066" is not. So we can draw at most a
partial truth-table for "j
because y".

jy

j
because y

T
T

?

T
F

F

F
T

F

F
F

F

The following is
also not a truth-functor:

Bill
Clinton was president of the USA and j.

This is because,
although the falsity of j is
enough to determine the truth-value of the whole, its truth is not. You,
of course, will know that the whole is true if the constituent is true.
But that is because you know some history. The fact that it is true is
not determined solely by the truth of the constituent and the meaning
of the sentence-functor.